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A generalized vertical coordinate transformation based on SPH(2) for efficient free surface flow simulations

This paper proposes three efficient particle methods for simulating free surface flows with complex bottom boundaries—BF-SPH, BFE-SPH, and $\sigma$ -SPH—that utilize a generalized Vertical Coordinate Transformation based on second-order accurate SPH(2) to accurately impose boundary conditions and optimize computational performance.

Original authors: Shujiro Fujioka, Kumpei Tsuji, Naoto Mitsume, Mitsuteru Asai

Published 2026-08-03

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CC BY 4.0

Original authors: Shujiro Fujioka, Kumpei Tsuji, Naoto Mitsume, Mitsuteru Asai

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to simulate a river flowing over a rocky riverbed, or a tsunami crashing onto a jagged coastline. To do this on a computer, scientists often use a technique called "particle methods." Instead of drawing a fixed grid of squares like a chessboard, they scatter thousands of tiny, invisible marbles (particles) through the water. These marbles move around, bump into each other, and carry information like speed and pressure. It's a bit like tracking a swarm of bees to understand how the whole hive moves. This approach is fantastic for messy, splashing water because the marbles can stretch and squeeze without getting stuck in a rigid grid.

However, there's a catch. To get an accurate picture of the water, these marbles usually need to be packed tightly and evenly everywhere. If the river is deep in the middle but shallow near the rocky banks, you still need a dense layer of marbles at the bottom of the deep part and a dense layer at the shallow part. This means you end up using way more marbles than necessary, which makes the computer calculation slow and expensive. It's like trying to take a high-resolution photo of a whole landscape, but you're forced to use the same number of pixels for the distant mountains as you do for the tiny pebbles right in front of your feet. The big question in this field is: Can we make the simulation smarter? Can we stretch our "marbles" so they are thin and wide in deep water, but short and fat in shallow water, without losing the accuracy of the picture?

This paper introduces a clever new way to solve that problem by inventing three "shape-shifting" tricks for these water particles. The researchers, led by Shujiro Fujioka and colleagues, propose a system called a "Vertical Coordinate Transformation." Think of this as a magical lens that changes the shape of the world the particles live in. Instead of forcing the particles to stay in a rigid box, the lens stretches or squashes the space itself so that the particles can be arranged more efficiently.

The first trick is the Bottom Boundary-Fitted (BF-SPH) method. Imagine the riverbed is a bumpy, wavy line. In a normal simulation, you have to pack particles tightly against every curve of that wavy line. This new method uses the lens to "flatten" the wavy riverbed into a straight, flat line in the computer's mind. The particles can now sit neatly in rows, making it much easier to calculate how they interact with the bottom, even if the real riverbed is full of rocks and curves.

The second trick, BFE-SPH, combines that flattening trick with a "squishing" trick. Here, the particles aren't just round marbles; they are allowed to become ellipsoids (like flattened eggs or rugby balls). In deep water, the particles can stretch out horizontally, becoming wide and flat. This means you need far fewer of them to cover the same area. It's like replacing a crowd of people standing shoulder-to-shoulder with a few people lying down in a row; you cover the same ground but use fewer bodies.

The third and most innovative trick is the σ-SPH method. This is the "smartest" of the bunch. It uses a coordinate system often found in oceanography (called the σ-coordinate) to automatically adjust the shape of the particles based on how deep the water is. In deep water, the particles stretch out wide to save space. As the water gets shallow near the shore, the particles automatically shrink back to their normal, rounder shape to keep the details sharp. It's like a camera that automatically zooms out when the scene is vast and zooms in when you need to see the small details, all without you touching a button.

To make sure these shape-shifting tricks don't ruin the math, the authors had to use a very precise calculation tool called SPH(2). Standard methods for these particle simulations are a bit like using a blurry lens; they work okay for simple things but get messy when you try to stretch the space. The SPH(2) method is a high-definition lens that can handle the complex math of stretching and squashing without losing accuracy. The researchers tested these new methods in several simulations, including a calm tank of water, a dam breaking over a triangular bump, and even a 3D simulation of water crashing against a wall.

The results were promising. In the simulations, the new methods successfully kept the water flowing correctly and maintained the right pressure, even when the particles were stretched into weird shapes. Most importantly, they proved that these tricks could drastically cut down the number of particles needed. In one test, the new "smart" method (σ-SPH) used only about one-quarter of the particles required by the standard method to get the same result. This means the computer could solve the problem roughly six times faster.

The paper also tackled a tricky side effect: when you stretch the space, the "volume" of each particle changes, which can mess up the total amount of water in the simulation. The authors developed a new correction technique to fix this, ensuring that the total volume of water stays constant even as the particles stretch and shrink. They found that with the right settings, the simulation could run for over 100 seconds without losing any water or gaining fake water.

In short, this paper suggests that by using these coordinate transformations and high-precision math, we can make computer simulations of water much faster and more efficient. It doesn't just work for calm water; it handles the chaos of breaking waves and complex riverbeds. While the authors note that these results come from computer simulations and that real-world testing is the next step, the findings suggest a powerful new way to model everything from tsunamis to coastal engineering without needing supercomputers to do the heavy lifting.

Technical Summary: Generalized Vertical Coordinate Transformation for Efficient Free Surface Flow Simulations

Problem Statement
Particle methods, such as Smoothed Particle Hydrodynamics (SPH) and Moving Particle Semi-implicit (MPS), are effective for simulating free surface flows with large deformations and moving discontinuities. However, they face significant computational cost challenges. Conventional particle methods require uniform particle spacing across the entire domain to maintain accuracy, necessitating a high number of particles even in regions where high resolution is not critical (e.g., deep offshore waters in tsunami simulations). While mesh-based methods can adjust grid sizes to reduce costs, traditional particle methods lack this flexibility. Furthermore, handling complex bottom boundaries (e.g., seabeds, riverbeds) often requires complex particle arrangements or boundary treatments that can degrade accuracy or stability. The authors identify a need for a method that improves computational efficiency by allowing non-uniform particle resolution (specifically in the vertical direction) and simplifying boundary handling without sacrificing the Lagrangian advantages of particle methods.

Methodology
The paper proposes a framework based on Generalized Vertical Coordinate Transformations (VCTs) applied to the SPH method. The core concept involves projecting the physical space (with complex boundaries and variable depth) onto a normalized "projected space" where the vertical coordinate is transformed. This allows for the use of ellipsoidal particles or simplified boundary representations in the projected space.

The implementation relies on three key technical components:

Coordinate Transformation: A transformation is defined where the physical vertical coordinate $z$ is mapped to a projected coordinate $b_z$ using scaling factors $\alpha(x,y)$ and height corrections $\beta(x,y)$ . This transformation requires the calculation of second-order derivatives, including cross-derivatives, to accurately solve the governing equations in the transformed space.
High-Accuracy Discretization (SPH(2)): To handle the complex derivative terms introduced by the coordinate transformation, the authors employ SPH(2), a second-order accurate SPH model. This model calculates second-order derivatives (including cross-derivatives) with high precision, which is essential for the stability and accuracy of the VCTs.
Volume Conservation: Since the coordinate transformation changes the Jacobian determinant (and thus the representative volume of particles) when $\alpha$ varies, a new volume conservation correction technique is introduced. This involves smoothing the Jacobian and adjusting particle volumes via a stabilization term in the pressure Poisson equation to prevent numerical instability and volume loss.

Three Proposed Methods
The authors generalize the VCT concept into three specific methods:

BF-SPH (Bottom Boundary-Fitted SPH): Inspired by body-fitted coordinate systems in finite difference methods. It transforms complex bottom boundaries into flat surfaces in the projected space ( $\beta = -h(x,y)$ , $\alpha=1$ ). This simplifies the imposition of bottom boundary conditions.
BFE-SPH (Bottom Boundary-Fitted Ellipsoidal SPH): Combines BF-SPH with an ellipsoidal particle model. It uses a constant vertical scaling factor ( $\alpha > 1$ ) to elongate particles vertically in the physical space, reducing the total particle count while maintaining the bottom-fitted advantage.
$\sigma$ -SPH: Utilizes a $\sigma$ -coordinate system (common in oceanography, e.g., Princeton Ocean Model) where the vertical scaling factor $\alpha$ varies based on water depth ( $\alpha = \bar{H} / (H - h(x,y))$ ). This allows for a stepwise transition in resolution, providing high resolution in shallow coastal areas and coarser resolution in deep offshore regions, significantly reducing the total number of particles.

Key Results
The methods were validated through 2-D and 3-D simulations of hydrostatic pressure, dam break problems, and dynamic flow over slopes:

Accuracy: The use of SPH(2) with VCTs significantly improved accuracy compared to standard ISPH. In hydrostatic tests, the $L_2$ error in pressure for SPH(2) was two orders of magnitude smaller than standard ISPH. The methods successfully reproduced hydrostatic pressure distributions even with complex bottom geometries and varying aspect ratios.
Efficiency: The $\sigma$ -SPH method demonstrated the highest efficiency. In a hydrostatic pressure problem, $\sigma$ -SPH reduced the particle count to approximately 1/4 and the computational time to 1/6 of the standard BF-SPH method, while maintaining comparable accuracy.
Stability and Volume Conservation: The proposed volume conservation technique (Condition C in the dynamic problem) successfully maintained total volume errors within $\pm 1\%$ over long simulation times (up to 100s), whereas methods without volume smoothing or with incorrect smoothing parameters exhibited volume loss or divergence.
3-D Capability: A 3-D dam break simulation confirmed that the $\sigma$ -SPH method could robustly handle complex 3-D flow phenomena, including wave run-up, reflection, and vortex formation, while maintaining low density errors and velocity divergence.

Significance and Claims
The paper claims that this work represents the first attempt to apply the $\sigma$ -coordinate system to a particle method. The authors emphasize that the success of these methods relies heavily on the use of SPH(2) to accurately compute the second-order derivatives required by the coordinate transformation.

The primary significance lies in the ability to:

Generalize VCTs for particle methods, bridging the gap between the flexibility of Lagrangian particle methods and the efficiency of coordinate-transformed mesh methods.
Dramatically reduce computational costs for oceanographic and coastal simulations (e.g., tsunamis) by allowing variable vertical resolution without compromising the accuracy of the free surface or boundary conditions.
Simplify boundary handling for complex bottom topographies by mapping them to flat surfaces in the computational domain.

The authors conclude that while the study primarily confirms the feasibility of these methods in 3-D, the $\sigma$ -SPH method offers a five-fold improvement in computational efficiency over non-biased methods, making it a promising tool for practical applications in coastal and marine engineering. They also note that the ability to control vertical resolution may offer future advantages in resolving near-wall turbulence, though this requires further investigation.

Technical Summary: Generalized Vertical Coordinate Transformation for Efficient Free Surface Flow Simulations

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